对称n-矩形的一些改进的高斯相关性不等式：扩展到一些多元伽马分布及进一步的概率不等式
Some improved Gaussian correlation inequalities for symmetrical n-rectangles extended to some multivariate gamma distributions and some further probability inequalities
AI总结：
本文改进了对称n-矩形的高斯相关性不等式，证明了MTP2条件下的联合分布适用性，并扩展至多元伽马分布，提出通过寻找逆为M矩阵的相关矩阵来推导可计算的概率不等式。
摘要：
如果绝对分量具有MTP2（二阶多元完全正）的联合累积分布函数（cdf），则对称n-矩形的高斯相关性不等式（GCI）会得到改进。此处给定类型的不等式至少适用于R^n或(0,无穷大)^n上所有具有处处正的光滑密度的MTP2-cdf。特别地，至少证明了某些具有任意正实“自由度”的无穷可分多元卡方分布函数（Krishnamoorthy和Parthasarathy意义上的伽马分布）是MTP2。此外，导出了广泛一类多元伽马分布的更多可数值计算的概率不等式，以及对GCI型不等式（以及具有三个而非两个分量组的类似类型不等式）在更特殊相关结构下的不同改进。这些不等式背后的主要思想是，对于给定的具有正相关的相关矩阵，寻找一个相关性更小的进一步相关矩阵，其逆是M矩阵，且相应的多元伽马分布函数在数值上是可用的。
英文摘要：
The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint cumulative distribution (cdf) which is MTP2 (multivariate totally positive of order 2). Inequalities of the here given type hold at least for all MTP2-cdfs on R^n or (0,infinity)^n with everywhere positive smooth densities. In particular, at least some infinitely divisible multivariate chi-square distribution functions (gamma distributions in the sense of Krishnamoorthy and Parthasarathy) with any positive real "degree of freedom" are shown to be MTP2. Moreover, further numerically calculable probability inequalities for a broad class of multivariate gamma distributions are derived and a different improvement for inequalities of the GCI-type - and of a similar type with three instead of two groups of components - with more special correlation structures. The main idea behind these inequalities is to find for a given correlation matrix with positive correlations a further correlation matrix with smaller correlations whose inverse is an M-matrix and where the corresponding multivariate gamma distribution function is numerically available.
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